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\begin{document}

{\bf  \large \noindent EXHIBITS}

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\bigskip


\begin{figure}[h]
%\begin{center}
\caption{Percentage CVaR contribution of asset 1 in function of its portfolio weight for a two-asset portfolio with asset returns that have a bivariate normal distribution with means $\mu_1$ and $\mu_2$, correlation $0.5$ and standard deviations $\sigma_1$ and $\sigma_2$, respectively.   }
\includegraphics[width=6.5cm,angle=270]{sensitivity_rho50.eps}
%\end{center}
\end{figure}


\begin{figure}[h]
%\begin{center}
\caption{Weight and CVaR allocation of bond-equity portfolios, together with the in-sample annualized mean and monthly 95\% CVaR over
the period January 1976-June 2010.     }
\vspace{1cm}
\scalebox{0.87}{
\begin{tabular}{|lc cc c cc c cc | } \hline
                          &  &	\multicolumn{2}{c}{Weight allocation} 	& &\multicolumn{2}{c}{CVaR allocation} & & Ann. mean & 95\% CVaR  \\
                         &  &	Bond & Equity 	& &Bond & Equity 	& &  & \\ \hline
Equal-weight	        &    &50\%  	&50\%       &	&3.47\%  & 96.53\% & & 8.90\% & 4.87\% \\
60/40 weight            &    &60\%  	&40\%       &  & 13.79\%     & 86.21\%   & & 8.63\% & 4.03\%\\
Min CVaR                &	 &96.86\%	& 3.14\%    &  & 96.86\%    & 3.14\%  & & 7.63\% & 2.44\%\\
Min CVaR concentration	&    &77.01\%	&22.91\%    &  & 50\%	     &50\%     & & 8.17\% & 3.00\%\\
60/40 risk allocation   &    &81.23\%	&18.77\%    &  & 60\%       & 40\%    & & 8.05\% & 2.80\%\\
 \hline
\end{tabular}
}
%\end{center}
\end{figure}

\vspace{-6cm}
%\begin{figure}[tb]
%\begin{center}
%\caption{Monthly statistics of real returns on total return indices of the Merrill Lynch US bond, S\&P500, MSCI EAFE and S\&P GSCI indices over the period January 1976 - December 2009.     }
%\vspace{1cm}
%\begin{tabular}{|lrrrr| } \hline
%&US bond &	S\&P 500	&EAFE&	 GSCI \\ \hline
%Mean (in \%) &	0.32	&0.52	&0.39&	0.10 \\
%StdDev	(in \%)&1.86	&4.46	&4.98&	5.50 \\
%Skewness&	0.89&	-0.74	&-0.78	&-1.03 \\
%Exc. Kurtosis	& 8.75	&2.48	&1.82&	5.45 \\
%95\% CVaR (in \%) &	1.24	&12.55&	13.51	&20.66 \\
% \hline
%\end{tabular}
%\end{center}
%\end{figure}

%\newpage

\begin{figure}[tb]
%\begin{center}
\caption{Annualized mean returns versus the monthly portfolio CVaR and largest percentage CVaR contribution for the mean-StdDev, mean-CVaR and mean-CVaR concentration efficient
portfolios. The frontier is  estimated using all January 1976-June 2010 monthly returns.  }
\includegraphics[width=12cm,angle=270]{frontier_fourassets.eps}
%\end{center}
\end{figure}


\begin{figure}[tb]
%\begin{center}
\caption{Weight and CVaR allocation of mean-StdDev, mean-CVaR and mean-CVaR concentration efficient
portfolios for various levels of annualized portfolio returns. The frontier is  estimated using all January 1976-June 2010 monthly returns.  }
\includegraphics[width=12cm,angle=270]{stackedweightsriskcont_efficientfrontier.eps}
%\end{center}
\end{figure}




\newpage

\begin{figure}[tb]
%\begin{center}
\caption{Stacked bar weight and CVaR contribution plots for the equal-weight, minimum CVaR and minimum CVaR concentration portfolios invested in the Merrill Lynch US bond, S\&P 500, MSCI EAFE and S\&P GSCI indices. The portfolios are rebalanced quarterly.}
\includegraphics[width=12cm,angle=270]{stackedweightsriskcont_benchmark.eps}
%\end{center}
\end{figure}

\newpage


\begin{figure}[tb]
%\begin{center}
\caption{Monthly CVaR of the equal-weight and risk budget optimized portfolios invested in the Merrill Lynch US bond, S\&P 500, MSCI EAFE and S\&P GSCI indices. The portfolios are rebalanced quarterly. The shaded regions indicate a bear market regime.   }
\includegraphics[width=12cm,angle=270]{portfolioCVaR.eps}
%\end{center}
\end{figure}

\newpage

\begin{figure}[tb]
%\begin{center}
\caption{Relative performance of the risk budget optimized portfolios versus the equal-weight portfolio invested in the Merrill Lynch US bond, S\&P 500, MSCI EAFE and S\&P GSCI indices. The portfolios are rebalanced quarterly. The shaded regions indicate a bear market regime.  }
\includegraphics[width=12cm,angle=270]{RelPerf_EW.eps}
%\end{center}
\end{figure}

\newpage

\begin{figure}[tb]
%\begin{center}

\caption{Summary statistics of monthly out-of-sample returns on investment strategies over the period January 1984 - June 2010.     }
\vspace{1cm}\scalebox{0.85}{
\begin{tabular}{|lc c cccc c cc| } \hline
                  &          &  	Equal             &    \multicolumn{4}{l}{Min CVaR}  & & \multicolumn{2}{l|}{Min CVaR Concentration} \\
                             \cline{4-7} \cline{9-10}
                       &     &   Weight            &      	        &   40\% Position   & 40\% CVaR 	     & ERC	       & &  & 40\% Position       \\
                    &        &    		          &                 &   Limit	   & Alloc Limit	         &             & &  & Limit 	       \\ \hline
 \multicolumn{10}{|l|}{\emph{Full period (in \%)}} \\
 \multicolumn{2}{|l}{Ann. Mean}   &           7.32    & 8.07      & 7.72	 & 7.99	 & 8.22         & & 8.23  & 7.63     \\
 \multicolumn{2}{|l}{Ann. StdDev} &          10.42    & 4.53      & 8.77	 & 5.35  & 5.78         & & 5.78  & 8.46    \\
 \multicolumn{2}{|l}{Monthly Hist CVaR} &     7.42    & 2.34      &	6.15     & 2.95  & 3.35         & & 3.35  & 5.91    \\
%HI of Hist $C_{i}$CVaR        &           0.06               & 0.21       &	0.09            & 0.16   & 0.12         & & 0.12  &  0.07      \\
 \multicolumn{2}{|l}{Portfolio turnover}    	  &           1.26        	     & 2.14	      &  3.55	        & 2.64	 & 1.74         & & 1.74  & 1.51    \\
 \multicolumn{10}{|l|}{\emph{Bear stock market (in \%)}} \\
 \multicolumn{2}{|l}{Ann. Mean  }    &        -24.36	& 6.31	      & -17.29	        & -0.61	 & -3.81         & & -3.79  & -16.51     \\
 \multicolumn{2}{|l}{Ann. StdDev}     &         15.46   & 6.00	      & 13.03	        & 7.10   & 7.44          & & 7.44  & 12.58     \\
 \multicolumn{2}{|l}{Monthly Hist  CVaR}   &	13.71   & 3.30        &	11.05            & 5.37  & 6.28          & & 6.27  & 10.80     \\
 \multicolumn{10}{|l|}{\emph{Normal/Bull stock market (in \%)}} \\
 \multicolumn{2}{|l}{Ann. Mean  } 	         &           13.37	    	    & 8.40	      & 12.50	        & 9.63	 & 10.52         & & 10.52  & 12.24      \\
 \multicolumn{2}{|l}{Ann. StdDev }       &   8.08    & 4.20	      & 6.92	        & 4.81   & 5.16          & & 5.16  & 6.67     \\
 \multicolumn{2}{|l}{Monthly Hist  CVaR}   &	        4.09                & 2.04        &	3.50            & 2.27   & 2.38         & & 2.38  & 3.38   \\
 \hline \multicolumn{10}{|l|}{ \emph{Drawdowns higher than 10\%}  }  \\
 \multicolumn{2}{|l}{Credit crisis$^{*}$}       &             0.48                & 0.09    	 & 0.37	           &0.13    & 0.15  	       & & 0.15 & 0.38            \\
 \multicolumn{2}{|l}{Dot-com bubble burst$^{**}$}&            0.25               &	        & 0.19             &	    &               & &       & 0.17              \\
 \multicolumn{2}{|l}{Asian-Russian crisis$^{***}$}&        0.12              &           &	               &	    &              & &      &      \\         \multicolumn{2}{|l}{ Black Monday$^{****}$}	    &            0.11            &              & 0.12	      	   &	   &          &       &          &
 \\	
 \hline \multicolumn{10}{|l|}{ \emph{Summary statistics on level and concentration of portfolio losses exceeding 10\%}  }  \\
$-w_t'r_t$ & \multicolumn{1}{l}{ median}   &        0.16             &  0.11      & 0.13             &0.12	    &  0.12        & & 0.12 & 0.12      \\
 &\multicolumn{1}{l}{  max  }    &        0.22             & 0.17    	& 0.21	           &0.15    & 0.14  	       & & 0.14 & 0.19            \\
$\max_i\frac{(w_{(i)t}r_{(i)t})}{w_t'r_t}$  & \multicolumn{1}{l}{ median}   &        0.43          &0.97	        & 0.51             &	 0.85   &   0.83            & & 0.83      & 0.58    \\
         & \multicolumn{1}{l}{ max  }    &        0.45          & 1.02    	 & 0.74	           &0.97    & 0.92  	       & & 0.92 & 0.66            \\  \hline
\end{tabular}
}
%\end{center}
{\scriptsize $^{*}$ May-Oct 2008 for the Min CVaR strategy,  June 2008-Feb 2009 for all other styles. $^{**}$ Sep 2000-Sep 2002.
\newline $^{***}$ April-Aug 1998. $^{****}$ Sep-Nov 1987. }
\end{figure}





\newpage

\begin{figure}[tb]
%\begin{center}
\caption{Stacked bar weight and CVaR contribution plots for the constrained minimum CVaR portfolios invested in the Merrill Lynch US bond, S\&P500, MSCI EAFE and S\&P GSCI indices. The portfolios are rebalanced quarterly.}
\includegraphics[width=12cm,angle=270]{MinCVaR_alternatives.eps}
%\end{center}
\end{figure}




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